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volume 6, issue 3, article 61, 2005.

Received 20 May, 2005;

accepted 01 June, 2005.

Communicated by:Th.M. Rassias

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Journal of Inequalities in Pure and Applied Mathematics

A NOTE ON CERTAIN INEQUALITIES FOR THE GAMMA FUNCTION

JÓZSEF SÁNDOR

Department of Mathematics and Informatics Babe¸s-Bolyai University

Str. Kog ˘alniceanu 1

400084 Cluj-Napoca, Romania.

EMail:[email protected]

c

2000Victoria University ISSN (electronic): 1443-5756 160-05

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A Note on Certain Inequalities for the Gamma Function

József Sándor

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Abstract

We obtain a new proof of a generalization of a double inequality on the Euler gamma function, obtained by C. Alsina and M. S. Tomás [1].

2000 Mathematics Subject Classification:33B15.

Key words: Euler gamma function, Digamma function.

1. Introduction

The Euler Gamma functionΓis defined forx >0by

Γ(x) = Z ∞

0

e^−tt^x−1dt.

By using a geometrical method, recently C. Alsina and M. S. Tomás [1] have proved the following double inequality:

Theorem 1.1. For allx∈[0,1], and all nonnegative integersnone has

(1.1) 1

n! ≤ Γ(1 +x)ⁿ Γ(1 +nx) ≤1.

While the interesting method of [1] is geometrical, we will show in what follows that, by certain simple analytical arguments it can be proved that (1.1) holds true for all real numbers n, and all x ∈ [0,1]. In fact, this will be a consequence of a monotonicity property.

Letψ(x) = ^Γ_Γ(x)⁰^(x) (x > 0)be the "digamma function". For properties of this function, as well as inequalities, or representation theorems, see e.g. [2], [4], [5], [7]. See also [3] and [6] for a survey of results on the gamma and related functions.

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2. Main Results

Our method is based on the following auxiliary result:

Lemma 2.1. For allx >0one has the series representation

(2.1) ψ(x) =−γ+ (x−1)

∞

X

k=0

1

(k+ 1)(x+k).

This is well-known. For proofs, see e.g. [4], [7].

Lemma 2.2. For allx >0, and alla≥1one has

(2.2) ψ(1 +ax)≥ψ(1 +x).

Proof. By (2.1) we can writeψ(1 +ax)≥ψ(1 +x)iff

−γ+ax

∞

X

k=0

1

(k+ 1)(1 +ax+k) ≥ −γ+x

∞

X

k=0

1

(k+ 1)(1 +x+k).

Now, remark that a

(k+ 1)(1 +ax+k) − 1

(k+ 1)(1 +x+k) = a−1

(1 +x+k)(1 +ax+k) ≥0

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J. Ineq. Pure and Appl. Math. 6(3) Art. 61, 2005

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Theorem 2.3. For alla≥1, the function

f(x) = Γ(1 +x)^a Γ(1 +ax)

is a decreasing function ofx≥0.

Proof. Let

g(x) = logf(x) =alog Γ(1 +x)−log Γ(1 +ax).

Since

g⁰(x) = a[ψ(1 +x)−ψ(1 +ax)],

by Lemma2.2 we getg⁰(x) ≤ 0, sog is decreasing. This implies the required monotonicity off.

Corollary 2.4. For alla≥1and allx∈[0,1]one has

(2.3) 1

Γ(1 +a) ≤ Γ(x+ 1)^a Γ(ax+ 1) ≤1.

Proof. Forx ∈(0,1], by Theorem2.3, f(1) ≤f(x)≤f(0), which byΓ(1) = Γ(2) = 1implies (2.3). Fora=n ≥1integer, this yields relation (1.1).

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References

[1] C. ALSINA AND M.S. TOMÁS, A geometrical proof of a new in- equality for the gamma function, J. Ineq. Pure Appl. Math., 6(2) (2005), Art.48. [ONLINE: http://jipam.vu.edu.au/article.

php?sid=517]

[2] E. ARTIN, The Gamma function, New York, Toronto, London, 1964.

[3] D.S. MITRINOVI ´C, Analytic Inequalities, Springer Verlag, 1970.

[4] A. NIKIFOROVANDV. OUVAROV, Éléments de la Théorie des Fonctions Spéciales, Ed. Mir, Moscou, 1976.

[5] J. SÁNDOR, Sur la fonction Gamma, Publ. C. Rech. Math. Pures Neuchâ- tel, Série I, 21 (1989), 4–7.

[6] J. SÁNDOR ANDD.S. MITRINOVI ´C (in coop. with B. CRSTICI), Hand- book of Number Theory, Kluwer Acad. Publ., 1996.

[7] E.T. WHITTAKER ANDG.N. WATSON, A Course of Modern Mathemat- ics, Camb. Univ. Press, 1969.